I wish to prove cos^2x-sin^2x=2cos^2x-1.

You can manipulate only the left side of the
equation.

For this reason, you'll use the Pythagorean
identity:

(sin x)^2 + (cos x)^2 =
1

(sin x)^2 = 1 - (cos
x)^2

You'll substitute (sin x)^2 to the left side of the
equation:

(cos x)^2 - [1 - (cos x)^2] = 2 (cos x)^2 -
1

We'll remove the brackets to the left
side:

(cos x)^2 - 1 + (cos x)^2 = 2 (cos x)^2 -
1

We'll combine like terms and we notice that LHS =
RHS:

2 (cos x)^2 - 1 = 2 (cos x)^2 -
1